The steady state of the inclined problem
Xiaoding Yang
TL;DR
The paper addresses the static equilibrium shapes of an open two‑dimensional fluid flowing over an inclined wall under a fixed contact angle, formulated as a volume‑constrained Euler–Lagrange problem for an energy combining gravity and surface tension. The authors develop a polar‑coordinate representation of the free boundary, apply a shift to eliminate the Lagrange multiplier, and construct solutions via a shooting method that tunes a single parameter (either the maximal height $u_m$ or the maximal slope angle $\psi_m$) to realize a prescribed volume $V$. A core result is the existence of steady states for any volume in the first case ($\psi_1\psi_2<0$), with uniqueness when $[[\gamma]]<0$, and more intricate existence statements in the second case ($\psi_1\psi_2\ge 0$) that depend on the geometry (including special subcases). The work provides a comprehensive framework for existence (and partial uniqueness) of equilibrium contact‑line configurations across all incline and contact settings, using variational principles, ODE shooting, and careful geometric analysis of boundary points and contact angles.
Abstract
The inclined problem is a problem that describes an open fluid flowing over an angled wall. It has broad applications in science and engineering. In this paper, we study the steady state of the inclined problem in two dimensions. The steady-state solution is depicted by the Euler-Lagrange equation of a given energy functional with a fixed contact angle as the boundary condition. By choosing a suitable maximal point to parameterize the surface of the fluid, we can construct a solution to this Euler-Lagrange equation via a shooting method in terms of the volume of the fluid. The construction works for any contact angle and any arbitrary inclined angle.